diff --git a/base/math.jl b/base/math.jl
index 47f9512d4efff..681cd07453def 100644
--- a/base/math.jl
+++ b/base/math.jl
@@ -1131,6 +1131,8 @@ julia> rem2pi(7pi/4, RoundDown)
 """
 function rem2pi end
 function rem2pi(x::Float64, ::RoundingMode{:Nearest})
+    isnan(x) && return NaN
+
     abs(x) < pi && return x
 
     n,y = rem_pio2_kernel(x)
@@ -1154,6 +1156,8 @@ function rem2pi(x::Float64, ::RoundingMode{:Nearest})
     end
 end
 function rem2pi(x::Float64, ::RoundingMode{:ToZero})
+    isnan(x) && return NaN
+
     ax = abs(x)
     ax <= 2*Float64(pi,RoundDown) && return x
 
@@ -1179,6 +1183,8 @@ function rem2pi(x::Float64, ::RoundingMode{:ToZero})
     copysign(z,x)
 end
 function rem2pi(x::Float64, ::RoundingMode{:Down})
+    isnan(x) && return NaN
+
     if x < pi4o2_h
         if x >= 0
             return x
@@ -1208,6 +1214,8 @@ function rem2pi(x::Float64, ::RoundingMode{:Down})
     end
 end
 function rem2pi(x::Float64, ::RoundingMode{:Up})
+    isnan(x) && return NaN
+
     if x > -pi4o2_h
         if x <= 0
             return x
diff --git a/test/numbers.jl b/test/numbers.jl
index 38c7c5c9b9e13..7520fbf3a2b2f 100644
--- a/test/numbers.jl
+++ b/test/numbers.jl
@@ -2611,6 +2611,13 @@ end
     @test rem2pi(T(-8), RoundUp)      ≈ -8+2pi
 end
 
+@testset "PR #36420 $T" for T in (Float16, Float32, Float64)
+    @test rem2pi(T(NaN), RoundToZero)  === T(NaN)
+    @test rem2pi(T(NaN), RoundNearest) === T(NaN)
+    @test rem2pi(T(NaN), RoundDown)    === T(NaN)
+    @test rem2pi(T(NaN), RoundUp)      === T(NaN)
+end
+
 import Base.^
 struct PR20530; end
 struct PR20889; x; end